AP Statistics Unit 7: Inference for Quantitative Data: Means Practice Exam

Assessment for Unit 7: Inference for Quantitative Data: Means

Estimated Time: 60 minutes

Exam Details

Questions: 43 total (43 multiple choice)
Time: 60 minutes (recommended)
Topics Covered: Unit 7: Inference for Quantitative Data: Means
Format: Multiple Choice questions matching real exam difficulty
Scoring: Instant results with detailed explanations

A. Multiple-Choice Questions (43 Questions)

Select the one best answer for each question.

1. A statistics student plans to construct a 95 percent confidence interval for a population mean based on a sample of size $n=10$. The population standard deviation is unknown. Which of the following best describes the shape of the sampling distribution of the $t$-statistic used for this interval compared to the standard normal distribution?

(A)The $t$-distribution is skewed to the right, whereas the standard normal distribution is symmetric.(B)The $t$-distribution has the same shape and spread as the standard normal distribution.(C)The $t$-distribution is symmetric but has a lower peak and thicker tails than the standard normal distribution.(D)The $t$-distribution is symmetric but has a higher peak and thinner tails than the standard normal distribution.

2. An environmental scientist wants to test whether the mean lead concentration in the soil of a specific region exceeds the safety limit of 400 ppm. The scientist collects a random sample of 25 soil specimens. The sample mean is 415 ppm with a sample standard deviation of 30 ppm. A histogram of the sample data shows a rough symmetry with no outliers. Which of the following is the correct expression for the test statistic $t$?

(A)$t = \frac{415 - 400}{30}$(B)$t = \frac{415 - 400}{\frac{30}{\sqrt{25}}}$(C)$t = \frac{415 - 400}{\sqrt{\frac{30}{25}}}$(D)$t = \frac{400 - 415}{\frac{30}{\sqrt{25}}}$

3. A researcher is studying the effectiveness of a new memory-enhancing program. Twenty participants take a memory test before the program and a similar version of the test after completing the program. The researcher calculates the difference in scores (Post - Pre) for each participant. The mean of the differences is 5.2 with a standard deviation of the differences of 4.0. Which of the following is the appropriate procedure to construct a 95 percent confidence interval for the mean improvement?

(A)A two-sample $t$-interval for the difference between means(B)A two-sample $z$-interval for the difference between means(C)A one-sample $t$-interval for a population mean(D)A one-sample $z$-interval for a population mean

4. A 95 percent confidence interval for the mean time (in minutes) it takes for a pain reliever to take effect is calculated to be $(18.5, 23.5)$. Which of the following is the correct interpretation of this interval?

(A)95 percent of the pain relief times in the sample are between 18.5 and 23.5 minutes.(B)There is a 0.95 probability that the sample mean time is between 18.5 and 23.5 minutes.(C)We are 95 percent confident that the true population mean time for the pain reliever to take effect is between 18.5 and 23.5 minutes.(D)If we were to take many samples, 95 percent of the sample means would fall between 18.5 and 23.5 minutes.

5. Researchers are comparing the durability of two types of fabric, Type A and Type B. They conduct a two-sample $t$-test to test the null hypothesis $H_0: \mu_A = \mu_B$ against the alternative $H_a: \mu_A \neq \mu_B$. The calculated $p$-value for the test is 0.03. Using a significance level of $\alpha = 0.05$, which of the following is the correct conclusion?

(A)Reject $H_0$; there is convincing evidence that the mean durability of Type A is different from Type B.(B)Reject $H_0$; there is convincing evidence that the mean durability of Type A is equal to Type B.(C)Fail to reject $H_0$; there is not convincing evidence that the mean durability of Type A is different from Type B.(D)Fail to reject $H_0$; there is convincing evidence that the mean durability of Type A is equal to Type B.

6. A university researcher investigates whether a new mindfulness program reduces stress levels in students. A random sample of 30 students is selected. Each student completes a stress assessment questionnaire before the program begins and completes the same questionnaire again after finishing the program. The researcher calculates the difference in scores (After $-$ Before) for each student. Which of the following is the most appropriate inference procedure to analyze the data?

(A)A two-sample $t$-test for the difference between means(B)A two-sample $z$-test for the difference between means(C)A one-sample $t$-test for a mean difference(D)A one-sample $z$-test for a population proportion

7. A factory produces steel bolts that are designed to have a mean length of 120 millimeters (mm). The quality control manager believes the machine is producing bolts that are too long. A random sample of 36 bolts is selected, resulting in a sample mean of $\bar{x} = 120.5$ mm and a sample standard deviation of $s = 1.2$ mm. What is the value of the test statistic for the hypothesis test $H_0: \mu = 120$ versus $H_a: \mu > 120$?

(A)$t = \frac{120.5 - 120}{1.2}$(B)$t = \frac{120.5 - 120}{\frac{1.2}{\sqrt{36}}}$(C)$t = \frac{120 - 120.5}{\frac{1.2}{\sqrt{36}}}$(D)$z = \frac{120.5 - 120}{\frac{1.2}{\sqrt{36}}}$

8. A 95% confidence interval for the mean weight of a specific breed of dog is calculated to be $(45.2, 48.8)$ pounds based on a random sample of dogs. Which of the following is the correct interpretation of the confidence level?

(A)There is a 95% probability that the mean weight of the sample is between 45.2 and 48.8 pounds.(B)There is a 95% probability that the true mean weight of this breed of dog is between 45.2 and 48.8 pounds.(C)If we were to take many random samples of the same size and construct a 95% confidence interval from each, approximately 95% of those intervals would contain the true mean weight of the breed.(D)Approximately 95% of all dogs of this breed weigh between 45.2 and 48.8 pounds.

9. A wildlife biologist wants to estimate the mean weight of a specific species of turtle in a wetland area. The biologist captures a random sample of 18 turtles and records their weights. The sample mean is 4.2 kg and the sample standard deviation is 0.8 kg. The population standard deviation is unknown. A dotplot of the sample data shows no strong skewness or outliers. Which of the following is the most appropriate procedure for constructing a confidence interval for the population mean weight?

(A)One-sample z-interval for a population mean(B)One-sample t-interval for a population mean(C)Two-sample t-interval for a difference between means(D)One-sample z-interval for a population proportion

10. A researcher is studying the fuel efficiency of a new car model. A random sample of 12 cars is tested, and the fuel efficiency (in miles per gallon) is recorded for each. The sample mean is 32 mpg and the sample standard deviation is 2.5 mpg. The researcher intends to construct a 95% confidence interval for the mean fuel efficiency. Under which of the following conditions would it be appropriate to calculate a one-sample t-interval?

(A)The sample size must be increased to at least 30 to satisfy the Central Limit Theorem.(B)The population of fuel efficiencies must be approximately normally distributed.(C)The sample standard deviation must be assumed to be equal to the population standard deviation.(D)The sample size is less than 10% of the population size, regardless of the shape of the population distribution.

11. Which of the following statements correctly compares a t-distribution with 5 degrees of freedom to the standard normal (z) distribution?

(A)The t-distribution has less area in the tails than the standard normal distribution.(B)The t-distribution has a higher peak at the mean than the standard normal distribution.(C)The t-distribution has more area in the tails and a lower peak than the standard normal distribution.(D)The t-distribution is skewed to the right, whereas the standard normal distribution is symmetric.

12. A manufacturing plant selects a random sample of 25 bolts to check their diameter. The sample mean diameter is 10.0 mm with a sample standard deviation of 0.2 mm. The plant manager wants to construct a 95% confidence interval for the true mean diameter of the bolts. The critical value t* for 24 degrees of freedom and 95% confidence is approximately 2.064. What is the margin of error for this confidence interval?

(A)0.008 mm(B)0.083 mm(C)0.413 mm(D)0.016 mm

13. A 90% confidence interval for the mean number of hours high school students sleep per night is calculated to be (6.5, 7.5). Which of the following is the correct interpretation of the 90% confidence level?

(A)There is a 90% probability that the true mean number of hours slept by all high school students is between 6.5 and 7.5.(B)90% of all high school students sleep between 6.5 and 7.5 hours per night.(C)If we took many random samples of the same size and constructed 90% confidence intervals for each, approximately 90% of those intervals would capture the true population mean.(D)The probability that the sample mean is between 6.5 and 7.5 is 0.90.

14. [Skill: 4.B | Topic: 7.3] A random sample of 50 students at a large university was selected to estimate the mean number of hours they study per week. A 95 percent confidence interval for the mean number of hours studied per week was calculated to be $(12.5, 16.5)$. Which of the following is the correct interpretation of the confidence interval?

(A)There is a 95 percent probability that the mean number of hours studied by all students at the university is between 12.5 and 16.5.(B)We are 95 percent confident that the mean number of hours studied by the 50 students in the sample is between 12.5 and 16.5.(C)We are 95 percent confident that the mean number of hours studied by all students at the university is between 12.5 and 16.5.(D)About 95 percent of all students at the university study between 12.5 and 16.5 hours per week.

15. [Skill: 4.D | Topic: 7.3] An environmental agency claims that the mean lead level in the water of a certain river is 15 parts per billion (ppb). To investigate this claim, scientists collected a random sample of water specimens and calculated a 99 percent confidence interval for the mean lead level to be $(16.2, 19.8)$ ppb. Based on the confidence interval, is the agency's claim supported?

(A)Yes, because the interval width is small, indicating high precision in the estimate.(B)Yes, because the confidence level is 99 percent, which is very high.(C)No, because the value 15 is not contained within the confidence interval.(D)No, because the midpoint of the interval is 18, which is greater than 15.

16. [Skill: 3.E | Topic: 7.3] A researcher plans to construct a 95 percent confidence interval for a population mean $\mu$. The pilot study results in a margin of error of 4.0. If the researcher wants to reduce the margin of error to 1.0 while maintaining the same 95 percent confidence level and assuming the population standard deviation remains constant, how must the sample size change?

(A)The sample size must be decreased by a factor of 4.(B)The sample size must be increased by a factor of 2.(C)The sample size must be increased by a factor of 4.(D)The sample size must be increased by a factor of 16.

17. [Skill: 4.A | Topic: 7.3] A biologist calculates a 90 percent confidence interval for the mean length of a certain species of fish using a sample of size $n=40$. If the biologist were to calculate a 99 percent confidence interval using the same sample data, how would the width of the new interval compare to the width of the original interval?

(A)The 99 percent interval would be narrower than the 90 percent interval.(B)The 99 percent interval would be wider than the 90 percent interval.(C)The width of the interval would remain the same, as the sample data has not changed.(D)The width cannot be compared without knowing the standard deviation of the sample.

18. [Skill: 4.D | Topic: 7.3] A study was conducted to determine if a new training program increases the mean number of sales made by employees. For each employee, the number of sales was recorded before and after the program, and the difference (After minus Before) was calculated. A 95 percent confidence interval for the mean difference $\mu_d$ was found to be $(-2.5, 5.1)$. Which of the following is the best justification for a conclusion regarding the effectiveness of the program?

(A)The program is effective because the upper bound of the interval (5.1) is positive and greater than the lower bound.(B)The program is effective because the interval includes positive values, suggesting sales increased.(C)There is not convincing evidence that the program is effective because the interval contains 0.(D)There is not convincing evidence that the program is effective because the interval contains negative values, which suggests sales decreased for everyone.

19. A consumer advocacy group suspects that a specific brand of cereal contains less than the advertised weight of 20 ounces per box. The group selects a random sample of 50 boxes and weighs the contents of each. Let $\mu$ represent the true mean weight of the cereal in all boxes of this brand. Which of the following are the appropriate null and alternative hypotheses for this investigation?

(A)$H_0: \mu = 20$ and $H_a: \mu < 20$(B)$H_0: \mu = 20$ and $H_a: \mu \neq 20$(C)$H_0: \bar{x} = 20$ and $H_a: \bar{x} < 20$(D)$H_0: \mu < 20$ and $H_a: \mu = 20$

20. A sleep researcher investigates whether a new relaxation technique reduces the time it takes for people to fall asleep. The researcher selects a random sample of 12 subjects. Each subject's time to fall asleep (in minutes) is measured on two consecutive nights: one night without the technique and one night using the technique. The order of the nights is randomized for each subject. The researcher plans to analyze the differences in time (Time without $-$ Time with). Which of the following is the most appropriate inference procedure for this study?

(A)A two-sample t-test for the difference between means(B)A matched pairs t-test for a mean difference(C)A two-sample z-test for the difference between means(D)A one-sample z-test for a population mean

21. An environmentalist wants to test if the mean pH level of a local stream is different from neutral (pH 7.0). She collects a random sample of 15 water specimens from various locations in the stream. A dotplot of the sample data reveals a distribution that is clearly skewed to the left with no outliers. Which of the following best describes the validity of the conditions for performing a one-sample t-test?

(A)The conditions are met because the sample size is less than 30.(B)The conditions are met because the t-test is robust to skewness for any sample size.(C)The conditions are not met because the sample size is small and the data are skewed.(D)The conditions are not met because the population standard deviation is unknown.

22. A high school track coach believes that a new shoe design helps athletes run the 100-meter dash faster. To test this, 10 runners run the 100-meter dash twice: once wearing the old shoes and once wearing the new shoes. The order of the shoes is randomized. Let $\mu_D$ be the true mean difference in finish times, defined as $\text{Time}_{new} - \text{Time}_{old}$. Which of the following pairs of hypotheses correctly tests the coach's belief?

(A)$H_0: \mu_D = 0$ and $H_a: \mu_D < 0$(B)$H_0: \mu_D = 0$ and $H_a: \mu_D > 0$(C)$H_0: \mu_D = 0$ and $H_a: \mu_D \neq 0$(D)$H_0: \bar{x}_D = 0$ and $H_a: \bar{x}_D < 0$

23. A city planner wishes to estimate the mean commute time for all 25,000 employed residents of a large city. The planner obtains a random sample of 3,000 employed residents and records their commute times. When checking conditions for constructing a confidence interval or conducting a significance test for the mean, which condition has most likely been violated?

(A)Random sampling condition(B)Normal/Large Sample condition(C)Independence (10% rule) condition(D)The condition that sigma is unknown

24. A manufacturer claims that the mean weight of a specific bag of chips is 50 grams. A quality control inspector suspects the filling machine is underfilling the bags. The inspector selects a random sample of 25 bags and finds a mean weight of 49.2 grams with a standard deviation of 1.5 grams. Which of the following is the correct test statistic for a test of the null hypothesis H_0: $\mu $= 50 versus the alternative hypothesis H_a: $\mu $< 50 ?

(A)t = $\frac{49.2 - 50}{1.5}$(B)t = $\frac{49.2 - 50}{\frac{1.5}{\sqrt{25}}}$(C)t = $\frac{50 - 49.2}{\frac{1.5}{\sqrt{25}}}$(D)t = $\frac{49.2 - 50}{\frac{1.5}{25}}$

25. A researcher conducts a significance test to determine if the mean time to complete a complex puzzle differs from the established average of 20 minutes. The hypotheses are H_0: $\mu $= 20 and H_a: $\mu \neq $20 . The test results in a p-value of 0.08. Which of the following is the correct interpretation of this p-value?

(A)There is a 0.08 probability that the null hypothesis is true.(B)There is a 0.08 probability that the mean time to complete the puzzle is different from 20 minutes.(C)Assuming the true mean time is 20 minutes, there is a 0.08 probability of obtaining a sample mean as extreme or more extreme than the one observed.(D)Assuming the true mean time is not 20 minutes, there is a 0.08 probability of obtaining the observed sample mean.

26. A city planner wants to determine if the mean daily commute time for residents has increased from the previous average of 35 minutes. A random sample of 40 residents yields a p-value of 0.032 for the test of H_0: $\mu $= 35 versus H_a: $\mu $> 35 . Using a significance level of $\alpha $= 0.05 , which of the following is the correct conclusion?

(A)Reject H_0 . There is convincing evidence that the mean daily commute time is equal to 35 minutes.(B)Reject H_0 . There is convincing evidence that the mean daily commute time is greater than 35 minutes.(C)Fail to reject H_0 . There is not convincing evidence that the mean daily commute time is greater than 35 minutes.(D)Accept H_a . The probability that the mean commute time is 35 minutes is 0.032.

27. A biologist is studying the length of a specific species of fish in a local lake. The biologist collects a random sample of 18 fish and calculates a mean length of 12.4 inches with a standard deviation of 2.1 inches. The biologist wishes to test if the mean length differs from 11 inches. What are the degrees of freedom and the specific t-distribution shape associated with the test statistic for this sample?

(A)df = 17; the distribution is skewed to the right.(B)df = 17; the distribution is symmetric and bell-shaped with heavier tails than a standard Normal distribution.(C)df = 18; the distribution is symmetric and bell-shaped with heavier tails than a standard Normal distribution.(D)df = 18; the distribution is exactly the same as the standard Normal distribution.

28. A standardized test is designed to have a mean score of 500. An administrator suspects the mean score of students at a specific school is different from 500. A random sample of students is taken, and the calculated t-statistic is t = -2.15 . Which of the following describes the area under the t-curve that corresponds to the p-value for this test?

(A)The area to the left of -2.15.(B)The area to the right of -2.15.(C)The area between -2.15 and 2.15.(D)The sum of the area to the left of -2.15 and the area to the right of 2.15.

29. A researcher wishes to estimate the difference in the mean weight of two different breeds of salmon, Breed A and Breed B. The researcher selects a random sample of 50 fish of Breed A and a separate independent random sample of 50 fish of Breed B. The weights are recorded, and the mean and standard deviation for each sample are calculated. Which of the following is the appropriate procedure to construct a confidence interval for the difference in the population mean weights?

(A)A two-sample z-interval for a difference between means(B)A two-sample t-interval for a difference between means(C)A paired t-interval for a mean difference(D)A two-sample t-interval for a difference between proportions

30. A researcher calculates a 99% confidence interval for the difference in mean cholesterol levels between two groups of patients, Group 1 and Group 2. The sample sizes are $n_1 = 15$ and $n_2 = 22$. Using the conservative method for degrees of freedom, what is the critical value $t^*$ that should be used for this interval?

(A)2.624 (based on df = 14)(B)2.807 (based on df = 21)(C)2.977 (based on df = 14)(D)2.724 (based on df = 35)

31. A 95% confidence interval for the difference between the mean hourly wage of workers in Industry A and Industry B ($\mu_A - \mu_B$) is calculated to be $(-1.50, 4.25)$ dollars. Based on this interval, which of the following is the best interpretation?

(A)We are 95% confident that the mean hourly wage in Industry A is between $1.50 less and $4.25 more than the mean hourly wage in Industry B.(B)We are 95% confident that the mean hourly wage in Industry A is greater than the mean hourly wage in Industry B.(C)There is a 0.95 probability that the true difference in mean hourly wages falls between -1.50 and 4.25.(D)Since the interval contains 0, we are 95% confident that the mean hourly wages in both industries are equal.

32. A botanist is comparing the growth of two varieties of tomato plants, Variety A and Variety B. A random sample of 30 plants of Variety A had a mean height of 45 cm, and a random sample of 30 plants of Variety B had a mean height of 42 cm. A 95% confidence interval for the difference in population mean heights ($\mu_A - \mu_B$) is calculated to be $(-1.5, 7.5)$. Based on this interval, which of the following is the correct conclusion regarding the claim that there is a difference in the mean heights of the two varieties?

(A)There is convincing evidence of a difference because the sample mean for Variety A is greater than the sample mean for Variety B.(B)There is convincing evidence of a difference because the interval contains more positive values than negative values.(C)There is not convincing evidence of a difference because the interval contains the value 0.(D)There is not convincing evidence of a difference because the interval width is too large to be precise.

33. A researcher calculates a 95% confidence interval for the difference between two population means, $\mu_1 - \mu_2$, using samples of size $n_1 = 40$ and $n_2 = 40$. The resulting interval is $(10, 30)$. Which of the following changes would most likely result in a narrower confidence interval, assuming the sample statistics (means and standard deviations) remain approximately the same?

(A)Decreasing the sample sizes to $n_1 = 20$ and $n_2 = 20$.(B)Increasing the confidence level to 99%.(C)Increasing the sample sizes to $n_1 = 100$ and $n_2 = 100$.(D)Using a critical value $t^*$ corresponding to a smaller number of degrees of freedom.

34. A statistics student constructs a 95% confidence interval for $\mu_{new} - \mu_{old}$, the difference in mean battery life (in hours) between a new battery type and an old battery type. The resulting interval is $(1.5, 4.2)$. Which of the following is the correct interpretation of this interval?

(A)There is a 0.95 probability that the difference in sample mean battery lives is between 1.5 and 4.2 hours.(B)95% of all batteries of the new type last between 1.5 and 4.2 hours longer than batteries of the old type.(C)We are 95% confident that the difference in the true mean battery life of the new type and the true mean battery life of the old type is between 1.5 and 4.2 hours.(D)We are 95% confident that the interval $(1.5, 4.2)$ contains the difference in the sample means.

35. A researcher wants to compare the average battery life of two different brands of smartphones, Brand A and Brand B. She randomly selects 40 phones of Brand A and 45 phones of Brand B. All phones are subjected to the same usage test until the battery dies. The usage times (in hours) are recorded. Which of the following is the most appropriate method for analyzing the data to determine if there is a significant difference in the mean battery life between the two brands?

(A)A one-sample t-test for a population mean(B)A matched pairs t-test for a mean difference(C)A two-sample t-test for the difference between two population means(D)A two-sample z-test for the difference between two population proportions

36. An education superintendent believes that the mean annual income of teachers in District X is greater than the mean annual income of teachers in District Y. She collects a random sample of teachers from each district to test this belief. Let $\mu_X$ represent the population mean income for District X and $\mu_Y$ represent the population mean income for District Y. Which of the following pairs of hypotheses is appropriate for this test?

(A)$H_0: \mu_X - \mu_Y = 0$ and $H_a: \mu_X - \mu_Y \neq 0$(B)$H_0: \mu_X - \mu_Y = 0$ and $H_a: \mu_X - \mu_Y > 0$(C)$H_0: \bar{x}_X - \bar{x}_Y = 0$ and $H_a: \bar{x}_X - \bar{x}_Y > 0$(D)$H_0: \mu_X - \mu_Y > 0$ and $H_a: \mu_X - \mu_Y = 0$

37. A study was conducted to compare the mean cholesterol levels of men and women in a large city. Random samples of 18 men and 20 women were selected. The data were plotted, and the dotplots for both samples showed no strong skewness and no outliers. Which of the following correctly explains why the Normal/Large Sample condition for a two-sample t-test is satisfied?

(A)Because the sample sizes are both less than 30, the condition is not satisfied.(B)Because the sum of the sample sizes ($18 + 20 = 38$) is greater than 30, the condition is satisfied.(C)Because the sample data show no strong skewness or outliers, the sampling distribution of the difference in sample means is approximately normal despite the small sample sizes.(D)Because the population distributions of cholesterol levels are known to be exactly normal.

38. A shoe manufacturer wants to test if a new rubber compound wears down slower than the current compound. They recruit 50 runners. For each runner, they flip a coin: if heads, the left shoe is made with the new compound and the right with the current; if tails, the left is current and right is new. After 3 months, the amount of tread wear is measured. The manufacturer plans to use a two-sample t-test for the difference of means. Is this the appropriate procedure?

(A)Yes, because there are two distinct groups of data (new compound and current compound).(B)Yes, because the sample size of 50 is large enough to satisfy the Normal condition.(C)No, because the data are paired by runner; a matched pairs t-test should be used.(D)No, because the random assignment was determined by a coin flip, which is not a valid randomization method.

39. A statistics student wants to compare the mean drying time of two different types of paint, Type A and Type B. The student paints 10 boards with Type A and 12 boards with Type B. The boards are identical, and the assignment of paint type to board is random. The mean drying time for Type A is $\bar{x}_A = 75.4$ minutes with a standard deviation of $s_A = 4.2$ minutes. The mean drying time for Type B is $\bar{x}_B = 72.8$ minutes with a standard deviation of $s_B = 5.1$ minutes. Which of the following is the correct expression for the test statistic to determine if there is a significant difference in the mean drying times?

(A)$t = \frac{75.4 - 72.8}{\sqrt{\frac{4.2}{10} + \frac{5.1}{12}}}$(B)$t = \frac{75.4 - 72.8}{\sqrt{\frac{4.2^2}{10} + \frac{5.1^2}{12}}}$(C)$t = \frac{75.4 - 72.8}{\frac{4.2}{\sqrt{10}} + \frac{5.1}{\sqrt{12}}}$(D)$t = \frac{75.4 - 72.8}{\sqrt{\frac{(4.2)^2}{10} + \frac{(5.1)^2}{12}}} \times \sqrt{20}$

40. A researcher conducts a two-sample t-test to determine if the mean cholesterol level of patients taking a new drug (Group 1) is lower than the mean cholesterol level of patients taking a placebo (Group 2). The hypotheses are $H_0: \mu_1 - \mu_2 = 0$ and $H_a: \mu_1 - \mu_2 < 0$. The calculated P-value for the test is 0.034. Which of the following is the correct interpretation of this P-value?

(A)There is a 0.034 probability that the null hypothesis is true.(B)There is a 0.034 probability that the alternative hypothesis is true.(C)Assuming the new drug has no effect compared to the placebo, there is a 0.034 probability of observing a difference in sample means as large or larger than the one observed in the direction of the alternative hypothesis.(D)Assuming the new drug is effective, there is a 0.034 probability of observing a difference in sample means as large or larger than the one observed.

41. An agricultural scientist wants to compare the yield of two corn varieties, Variety X and Variety Y. Independent random samples of 15 plots for Variety X and 15 plots for Variety Y are planted. The scientist tests $H_0: \mu_X - \mu_Y = 0$ against $H_a: \mu_X - \mu_Y \neq 0$. The resulting P-value is 0.08. Using a significance level of $\alpha = 0.05$, which of the following is the correct conclusion?

(A)Reject $H_0$. There is convincing evidence that the mean yields of the two varieties are different.(B)Reject $H_0$. There is not convincing evidence that the mean yields of the two varieties are different.(C)Fail to reject $H_0$. There is convincing evidence that the mean yields of the two varieties are the same.(D)Fail to reject $H_0$. There is not convincing evidence that the mean yields of the two varieties are different.

42. A school district is comparing the mean scores on a standardized math test for two different high schools, High School A and High School B. A random sample of 12 students from High School A and 15 students from High School B is selected. The dotplots of the sample data show that the distribution for High School A is roughly symmetric, but the distribution for High School B is strongly skewed to the left with a potential low outlier. Can a two-sample t-test for the difference in means be safely performed?

(A)Yes, because the sample sizes are both less than 30.(B)Yes, because t-procedures are robust against non-normality regardless of sample size.(C)No, because the samples are not independent.(D)No, because the sample sizes are small ($n < 30$) and one of the sample distributions suggests the population may not be Normal.

43. A study compared the fuel efficiency (in miles per gallon) of two types of cars, Hybrids and Standard Sedans. Hybrid: $n=40$, $\bar{x}=48$, $s=5.2$ Sedan: $n=40$, $\bar{x}=32$, $s=4.8$ The researcher tests the hypothesis that Hybrids have a higher mean mpg than Sedans ($H_a: \mu_H > \mu_S$). The test statistic is calculated to be $t \approx 14.3$. Which of the following statements best justifies the conclusion at $\alpha = 0.01$?

(A)The P-value is approximately 0, which is less than 0.01. Thus, we reject $H_0$ and conclude Hybrids have a higher mean mpg.(B)The P-value is approximately 1, which is greater than 0.01. Thus, we fail to reject $H_0$.(C)The test statistic is large, but because the sample sizes are equal, we cannot draw a conclusion.(D)We reject $H_0$ because the difference in means ($48-32=16$) is larger than the standard deviation ($5.2$).

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